{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:NQCCYWPESHPAEZ2IUI3R7FFY6N","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"cd63724412441b166e38f0562156e5e29de20b04eb2861d910a0c4a74864c2e4","cross_cats_sorted":["cs.LG","math.PR","stat.ML","stat.TH"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2025-08-01T13:25:59Z","title_canon_sha256":"b7a8b454c21ea4128bbcb0c0d7f314c051d1dace8f13b81ddeb584e98c9087bb"},"schema_version":"1.0","source":{"id":"2508.00617","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2508.00617","created_at":"2026-07-05T11:46:59Z"},{"alias_kind":"arxiv_version","alias_value":"2508.00617v1","created_at":"2026-07-05T11:46:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.00617","created_at":"2026-07-05T11:46:59Z"},{"alias_kind":"pith_short_12","alias_value":"NQCCYWPESHPA","created_at":"2026-07-05T11:46:59Z"},{"alias_kind":"pith_short_16","alias_value":"NQCCYWPESHPAEZ2I","created_at":"2026-07-05T11:46:59Z"},{"alias_kind":"pith_short_8","alias_value":"NQCCYWPE","created_at":"2026-07-05T11:46:59Z"}],"graph_snapshots":[{"event_id":"sha256:bfe8e132ec2098aa053643bf483e894d1ecf6cdb11a4296d8f6686d0b226fcec","target":"graph","created_at":"2026-07-05T11:46:59Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2508.00617/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Conditioning, the central operation in Bayesian statistics, is formalised by the notion of disintegration of measures. However, due to the implicit nature of their definition, constructing disintegrations is often difficult. A folklore result in machine learning conflates the construction of a disintegration with the restriction of probability density functions onto the subset of events that are consistent with a given observation. We provide a comprehensive set of mathematical tools which can be used to construct disintegrations and apply these to find densities of disintegrations on differen","authors_text":"Jon Cockayne, Marvin Pf\\\"ortner, Natha\\\"el Da Costa","cross_cats":["cs.LG","math.PR","stat.ML","stat.TH"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2025-08-01T13:25:59Z","title":"Constructive Disintegration and Conditional Modes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.00617","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b943e5d95470e42710cf92c1673b148cd18b49ccdffa7f994ef91f07e4236601","target":"record","created_at":"2026-07-05T11:46:59Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"cd63724412441b166e38f0562156e5e29de20b04eb2861d910a0c4a74864c2e4","cross_cats_sorted":["cs.LG","math.PR","stat.ML","stat.TH"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2025-08-01T13:25:59Z","title_canon_sha256":"b7a8b454c21ea4128bbcb0c0d7f314c051d1dace8f13b81ddeb584e98c9087bb"},"schema_version":"1.0","source":{"id":"2508.00617","kind":"arxiv","version":1}},"canonical_sha256":"6c042c59e491de026748a2371f94b8f35df513c032bcfa37f8e824b9c65eb9be","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6c042c59e491de026748a2371f94b8f35df513c032bcfa37f8e824b9c65eb9be","first_computed_at":"2026-07-05T11:46:59.070016Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:46:59.070016Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WctJlQT9qGHYqQUs3imds5pSQlj11igZpA79xT5b+06cJOtvCJig+XcpVRcmL8ZcDhuFYa/Ts9+83u48eYjjDg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:46:59.070505Z","signed_message":"canonical_sha256_bytes"},"source_id":"2508.00617","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b943e5d95470e42710cf92c1673b148cd18b49ccdffa7f994ef91f07e4236601","sha256:bfe8e132ec2098aa053643bf483e894d1ecf6cdb11a4296d8f6686d0b226fcec"],"state_sha256":"e1085cde1fa8f9626a8d85cf1244812e57d139829346097ba121a168288b0a41"}